Feasible alphabets for communicating the sum of sources over a network
ISIT'09 Proceedings of the 2009 IEEE international conference on Symposium on Information Theory - Volume 2
Algebraic network coding: a new perspective
ISIT'09 Proceedings of the 2009 IEEE international conference on Symposium on Information Theory - Volume 1
Cross-layer optimizations for intersession network coding on practical 2-hop relay networks
Asilomar'09 Proceedings of the 43rd Asilomar conference on Signals, systems and computers
Path gain algebraic formulation for the scalar linear network coding problem
IEEE Transactions on Information Theory
| Hi-index | 754.90 |
If beta and gamma are nonnegative integers and F is a field, then a polynomial collection {p1,hellip ,Pbeta} sube Z[alpha1,hellip, alphagamma] is said to be solvable over F if there exist omega1hellip, omegagamma isin F such that for all i = 1,hellip, beta we have pi(omega1hellip, omegagamma) = 0. We say that a network and a polynomial collection are solvably equivalent if for each field F the network has a scalar-linear solution over F if and only if the polynomial collection is solvable over F. Koetter and Medard's work implies that for any directed acyclic network, there exists a solvably equivalent polynomial collection. We provide the converse result, namely, that for any polynomial collection there exists a solvably equivalent directed acyclic network. (Hence, the problems of network scalar-linear solvability and polynomial collection solvability have the same complexity.) The construction of the network is modeled on a matroid construction using finite projective planes, due to MacLane in 1936. A set psi of prime numbers is a set of characteristics of a network if for every q isin psi, the network has a scalar-linear solution over some finite field with characteristic q and does not have a scalar-linear solution over any finite field whose characteristic lies outside of psi. We show that a collection of primes is a set of characteristics of some network if and only if the collection is finite or co-finite. Two networks N and N' are Is-equivalent if for any finite field F, N is scalar-linearly solvable over F if and only if N' is scalar- linearly solvable over F. We further show that every network is ls-equivalent to a multiple-unicast matroidal network.